Decomposing integrated circuit layout

ABSTRACT

Various embodiments of the invention provide techniques to ensure a layout for an integrated circuit is split-able. In a method embodiment, a layout is generated in a customer site having a layout library as inputs wherein the library provides exemplary layouts that have been verified to be spit-able and that can be used and layouts that can cause conflicts to avoid. A real-time odd cycle checker is also provided in which the checker identifies in real time conflict areas and odd cycles as they arise during layout generation. To reduce memory usage layouts of various devices may be separated so that each individual layout or a small number of layouts, rather than a large layout for the whole application circuit, can be checked against conflicts. Once the layout is ready at the customer site, it is sent to the foundry site to be decomposed into two masks and taped-out. Other embodiments are also disclosed.

TECHNICAL FIELD

Embodiments of the invention are generally related to integrated circuitlayouts. In various embodiments, mechanisms are provided to enablelayouts meeting the double patterning technology requirements includingsplitting the layouts into different masks.

BACKGROUND

Integrated circuit layouts cannot be split into two masks if theyinclude conflict cycles. A conflict cycle may be referred to as an oddcycle because it is a cycle in the conflict graph that contains an oddnumber of edges. Many layout designers (e.g., customers of asemiconductor foundry) do not have tools to check those conflict cyclesand thus can violate the rules to split the layouts. Various layoutapproaches cannot fix an odd cycle and associated problems because ofthe constraint on the stitch locations. A straightforward method is tosplit the patterns, connect the polygons by stitch areas, and use amatrix global solver to decompose the layout. In that method, however,memory consumption is huge; cycle time increases; customer generally isnot involved in the layout creation process; and the layout creationprocess is not user-friendly because different approaches are employedto decompose the layout when the polygons are cut and not cut.

BRIEF DESCRIPTION OF THE DRAWINGS

The details of one or more embodiments of the invention are set forth inthe accompanying drawings and the description below. Other features andadvantages of the invention will be apparent from the description,drawings, and claims.

FIG. 1 shows a flowchart illustrating a method to generate a layout andsplit it into two masks, in accordance with an embodiment of theinvention.

FIG. 2 shows exemplary layouts that have been verified to be split-ableand thus layout conflicts can be avoided, in accordance with anembodiment.

FIG. 3 shows a flowchart illustrating a method to generate a layout andcheck for odd cycles in real-time, in accordance with an embodiment.

FIG. 4 shows exemplary stitching elements in accordance with anembodiment.

FIG. 5 shows diagrams illustrating pre-stitching, in accordance with anembodiment.

FIG. 6 shows a diagram illustrating guard banding or boxing inaccordance with an embodiment.

FIG. 7 shows a flowchart illustrating a method to decompose a layoutthat was previously created.

FIG. 8 shows graphs illustrating stitch optimization with two exemplarycolor sets, in accordance with an embodiment.

FIG. 9 shows graphs illustrating stitch optimization having exemplaryfour color sets, in accordance with an embodiment.

FIG. 10 shows a flowchart illustrating a method to optimize stitchelements or the number of polygons to be cut, in accordance with anembodiment.

FIG. 11 shows graphs illustrating merging color sets, in accordance withan embodiment.

FIG. 12 shows graphs illustrating terms used in this document.

Like reference symbols in the various drawings indicate like elements.

DETAILED DESCRIPTION

Embodiments, or examples, of the invention illustrated in the drawingsare now being described using specific language. It will nevertheless beunderstood that no limitation of the scope of the invention is therebyintended. Any alterations and modifications in the describedembodiments, and any further applications of principles of the inventiondescribed in this document are contemplated as would normally occur toone skilled in the art to which the invention relates. Reference numbersmay be repeated throughout the embodiments, but this does notnecessarily require that feature(s) of one embodiment apply to anotherembodiment, even if they share the same reference number.

Exemplary Method Embodiment to Decompose a Layout

FIG. 1 shows a flowchart illustrating a method embodiment decomposing anintegrated circuit layout into two masks. In this illustration, thefoundry provides a solution for customers to make sure the layout can besplit into two masks, e.g., a mask A and a mask B. Further, the customer(e.g., a customer layout designer) generates a first part of the layoutat the customer site (e.g., blocks 110 to 120) while the foundry (e.g.,a foundry layout designer) generates a second part of the layout at thefoundry site (e.g., blocks 125 to 140). Additionally, the foundrypreviously provided the double patterning technology (DPT) library, theDPT guidelines, the conflict checker, and the guard band or boxingguidelines to the customer so that these tools are available to be usedin generating the layout.

In block 110 the customer starts generating a layout. In block 112 a DPTlibrary and/or DPT guidelines are available to help avoiding conflictcycles (e.g., odd cycles) so that the finalized layout may be split-ableinto two masks. Generally, the library includes exemplary layouts thathave been verified to be split-able and may be used during layout. Thelibrary also includes layouts that violate the layout split rules to beavoided.

In block 115 the customer uses a real-time odd cycle checker and fixerpreviously provided to check and, if appropriate, fix the odd cycles.Depending on implementations, in some embodiments, the checker and thefixer may be two distinct or integrated packages of software.Alternatively, the fixer may be part of the checker.

In block 117, to reduce memory usage, the customer may “guard band” or“box” the devices, i.e., keeping layouts of individual devices orsmaller groups of devices in layout units so that these layout units,rather than the layout of a whole application circuit, may be checkedagainst conflicts. In an embodiment, if guard band or box is used, theinformation is saved (e.g., the layouts or guard bands are marked withlayer numbers) and can be used later. For example, the decomposer inblock 130 reads the layer numbers and acts accordingly.

In block 120 the customer finishes the layout at the customer site.

In block 125 the customer layout is ready for decomposition at thefoundry site.

In block 130 the foundry uses a layout decomposition tool to decomposethe layout, e.g., split it into two masks.

In block 135, the layout results in two masks, e.g., mask A and mask B.

In block 140, the layout including the two masks A and B, is used intaping-out.

Library and Guidelines to Avoid Conflict Cycles

FIG. 2 shows exemplary layouts in a layout library. These layoutsinclude layouts that have been verified to be split-able and thus may beused in generating a layout for a particular application/circuit, andlayouts that can cause conflicts, e.g., not split-able, to be avoided.For example, layout 210 is not split-able because there are six conflictedges resulting from the triangle inequality of the 2 conflict edgesshown. Layout 220 is split-able because, even though a loop exists, thecycle is even (e.g., the number of conflict edges is six). Layout 230 isnot split-able because a loop exists and the number of conflict edges orthe conflict cycle is odd (e.g., the number of conflict edges is three).Both layouts 240 and 250 show a distance L at different locations, andthe layout should not be used if a distance L at a particular locationis less than a predetermined value (e.g., 20 nm, 30 nm, 40 nm, etc.,depending on technologies) and polygon cutting is not allowable becausewithout polygon cutting, the layout is not split-able. Self-conflictlayout (e.g., the U shape pattern in layout 250) is also not split-ableif polygon cutting is not allowable. In situations where polygon cuttingis allowable and the conflict can be resolved, layouts 240 and 250 maybe used. The separation space between two polygons varies for differenttechnologies and may be different at different locations, including, forexample, separation space between end-end, end-run, corner-corner, etc.Depending on applications the library may be integrated into thesoftware package (e.g., the checker, the decomposer, etc.) and/oraccessible during layout generation.

Check and Fix Conflict Cycles in Real-Time

FIG. 3 shows a flowchart 300 illustrating a method embodiment forchecking a conflict cycle in real time, e.g., block 115 in FIG. 1. In anembodiment, while the layout is generated, a checker software tool(e.g., the “checker”) in real time identifies the conflict polygons, theconflict areas, the conflict edges, the odd cycles, etc., if they arise.For example, the checker provides an arrow (e.g., a red arrow) betweentwo conflict areas or polygons when the separation space between twopolygons is smaller than a required or predetermined space. When theconflict edges form a loop, the checker counts the number of edges andhighlights the edges if they are odd, i.e., they form an odd and thusconflict cycle.

In block 305 the layout designer generates (e.g., draws) a layoutcorresponding to a circuit, a system, an application, etc. During layoutgeneration polygons representing circuits (e.g., sub circuits) areformed.

In block 310 the checker applies the split rules against the layoutbeing generated. In an embodiment, rules regarding layout split werepreviously provided to the checker. Generally, the checker determines ifsplitting is specified by the designer and/or required (e.g., to meetsome specifications). Depending on applications a layout is to be splitif the space between two polygons is too small, e.g., less than apredetermined value. Further, the predetermined space may be defined bythe distance between the edges or the center of the polygons, which maybe referred to as a pitch. When splitting is specified and/or required(and if the splitting is allowed), the checker determines if the layoutis split-able. In an embodiment, a layout is split-able into two masksif there is no odd cycle in the layout. Further, during layoutgeneration, if the designer forms an odd cycle, the checker providesindications that an odd cycle was formed.

In block 312, the checker determines whether splitting is necessary whenthere is a conflict edge. If splitting is not necessary, the checker inblock 315 determines whether the designer finishes drawing the layout.If the designer finishes drawing the layout, the checker in block 320provides the output layout. During layout generation, if pre-stitchingwas performed (e.g., in block 330), the checker in block 320 saves thelocation of the pre-stitching areas to a file or one or more layers andmakes them available as appropriate.

If in block 315, the designer has not finished with the layout, thedesigner continues generating the layout. That is, the method embodimentflows to block 305, and the designer continues drawing the layout goingthrough the flow of flowchart 300.

In block 322, the checker determines if polygon cutting is allowed. Inan embodiment, the designer, through an input mechanism (e.g., aprogramming parameter, a graphic user interface, etc.), specifies ifpolygon cutting is allowed. Depending on applications, the checker mayquery whether the designer wanted polygon cutting or not, and thedesigner may respond through such a mechanism.

If polygon cutting is not necessary for the layout to be split-able, thechecker in block 340 identifies conflict polygons, and, in block 345,identifies conflict edges. In block 350, the checker checks for oddcycles.

In block 352 the checker determines if there is any odd cycle in thelayout, and if there is not any odd cycle, the method embodiment flowsto block 315 and continues therefrom.

If in block 352, the checker determines an odd cycle exists, the checkerin block 355 attempts to fix the odd cycle. Depending on applications,the checker may invoke an odd cycle fixer, which could be part of thechecker, a package of software integrated into the checker, or anindependent package of software, etc.

In block 360, the checker determines if vertical integrated resolutionenhancement technology (VIRET) could be used. VIRET, in an embodiment,is a software package to enlarge the margin of the stitching area. Thoseskilled in the art will recognize that when a margin of a stitching areais enlarged, the corresponding conflict edge, conflict polygon and thusodd cycle may be resolved. In an embodiment this information is saved toa file or layers.

In block 365, the checker determines whether the designer wanted to useVIRET. In many situations, the design may seek to not use VIRET, e.g.,not to enlarge the margin for the stitching area. If VIRET is used,e.g., after the margin has been enlarged, the checker (re)-identifiesthe conflict edges, and the method flows to block 345 and continuestherefrom.

If in block 365, the checker determines that VIRET is not used, themethod flow proceeds to block 370 where the designer modifies his/herlayout. Once the layout is modified, the method embodiment flows toblock 310 and continues therefrom.

In block 322, however, if the checker determines if the polygon cuttingis allowed, then the checker, in block 325, based on the polygons havingbeen created, identifies the conflict and connection areas.

In block 330, the checker performs pre-stitching to identify locationcandidates for stitching. Here, like other functions, the checker may ormay not include internal pre-stitching capabilities. In any event, thechecker may invoke some pattern matching or recognition software packageto identify the pre-stitching locations.

In block 332 the checker, based on the pre-stitching information,determines whether a location is feasible for stitching. If stitching isfeasible, the checker identifies the conflict edges in block 345, andthe method flows continues therefrom. If stitching is not feasible inblock 332, then the checker in block 335 removes the correspondingstitching location candidate. In effect, the checker merges the polygonportions separated by this stitching location candidate and thus reformsa conflict area if appropriate. The method flows to and continues fromblock 345.

FIG. 3 is illustrated in the context of using the conflict checker inreal-time. That is, the checker is functioning while a layout designeris drawing the layout. In this aspect many conflict check rules (e.g.,the minimum space requirement between two polygons, the odd cycleviolations, etc.) are applied, and the conflicts (e.g., conflict edges,conflict polygons, odd cycles, etc.) are displayed graphically in realtime. In some embodiments, a “button click” approach is adopted todetermine if an existing layout is split-able or needs modification. Forexample, an input layout (e.g., a layout that was previously drawn) isprovided as an input to the checker, and appropriate parameters are setso that the method embodiment may flow as desire, and if an odd cycleexists in the input layout, the checker can provide a message indicatingmodification to the layout is needed. In this example, the methodembodiment flows through blocks 305, 310, 312, 322, 325, 330, 332, 335,345, 350, 352, 355, 360, 365, and 370. In this flow, the parameterscorresponding to splitting (e.g., block 312), polygon cutting (e.g.,block 322), applying VIRET (e.g., block 360) are set to yes (Y), yes(Y), and no (N), respectively. Further, the checker recognizespre-stitching is feasible (e.g., block 332) and an odd cycle (e.g.,block 352) is detected. For another flow, appropriate parameters are setaccordingly.

Pre-Stitching

Pre-stitching (e.g., block 330) identifies candidate locations forstitching. Pre-stitching checks to determine if a stitching element fitsin a connection area. FIG. 4 shows exemplary stitching elementsincluding a bar 405, a square 410, an L-shape 415, a Z shape 420, aT-shape 425, and a cross-shape 430. Stitching elements in FIG. 4 are forillustration, embodiments of the invention are not limited to thoseelements, but are applicable to other stitching elements of differentshapes and sizes including, for example, composites and combinations ofthe shapes shown in FIG. 4. Further, orientation of a stitching elementcould be any angle. Pre-stitching locations may or may not be the finalstitching locations. Pre-stitching locations may be user-defined and/ordetermined by available software that uses pattern matching to match thestitching elements (e.g., those in FIG. 4) to the patterns in theconnection areas potentially available for stitching. Depending onapplications, the software may move the stitching elements (e.g., up,down, right, left, etc.) to pattern match. Generally, when a connectionarea and a stitching element have the same shape, a location is selectedfor stitching if the stitching element is smaller than the connectionarea. Based on the potential stitching locations, embodiments of theinvention can resolve odd cycles. In some embodiments, the disclosedmethod is advantageous over other approaches that use pre-cuttingbecause, depending on applications, pre-cutting can still result in oddcycles, while pre-stitching in various embodiments of the invention canhelp resolve the odd cycles. For example, when stitching is feasible,embodiments can reduce the number of conflict edges or remove a cycle,and thus resolve the odd cycle (e.g., changing from an odd cycle to aneven cycle or to a non-cycle).

In various embodiments of the invention, when a stitching locationcandidate is not feasible for stitching (non-stitching location), thechecker removes that non-stitching location. In effect, the checkermerges the portions of the polygons separated by the non-stitchinglocation. FIG. 5 shows diagrams illustrating pre-stitching in accordancewith an embodiment. In diagram 510, two exemplary conflict areas 510-1-1and 510-1-2 are shown, two connection areas (e.g., locations) 510-1 and510-2 have been identified, and two stitching elements 405 and 415(e.g., based on pattern matching or recognition) are used to compareagainst locations 510-1 and 510-2 respectively. For further illustrationpurposes, stitching element 405 does not fit in area 510-1 whilestitching element 415 fits in area 510-2. As a result, location 510-2 isidentified as a stitching location candidate, and the size of stitchingelement 415 is used for stitching. In contrast, area 510-1 is notconsidered a stitching location candidate, and, consequently,embodiments of the invention remove the connection area 510-1 or mergethe polygons 510-1-1 and 510-1-2. Diagram 520 illustrates the result ofpre-stitching diagram 510, showing stitching element at location 510-2as a stitching location candidate and conflict areas 510-1-1 and 510-1-2have been merged once location 510-1 is no longer a stitching locationcandidate. Further, diagram 520 does not include an odd cycle ascompared to diagram 510 that includes an odd cycle if polygon cutting isnot allowable.

Fixer

Some embodiments of the invention provide fixing guidelines that includeenlarging space between polygons and/or apply VIRET (vertical integratedresolution enhancement technology) solution to extend the stitching areamargin so that the stitching area is as small as possible. Once themargin is extended (e.g., enlarged), the conflict space is resolved.Depending on situations, the space between two conflict polygons stillmay not meet the minimum space requirement but is allowed if VIRET isused. As a result, the odd cycle is resolved. In various embodiments,VIRET is applicable for metal layers, and VIRET takes two or more inputlayers. Actually, VIRET can be viewed as another stitching but isgenerated at another layer (e.g., a via layer). Further, if there arefinal contours in the masks that can cause bridging and necking risks(e.g., hot spots), VIRET may be used or designers have to pull back theline-end (e.g., the space is enlarged) and thus reduces this risk, butthe cell size may increase. In an embodiment, where there are a primarymask and a secondary mask, a dummy via having a corresponding pattern ora pattern modified from the original pattern is put at a via layer toprovide enough coverage for each mask such that after the Dual Damasceneprocess the dummy via can link the two metal lines. The minimum area ofcoverage or overlap between the dummy or the modified pattern and themetal pattern in each mask depends on process capability and/or circuitperformance, but the pattern modification is subject to the design rulesfor the particular technology (e.g., process).

Guard Band or Boxing

Some embodiments of the invention provide guard bands or boxes to reducememory usage when running a software package (e.g., the conflictchecker, the fixer, the decomposer etc.). These embodiments limit thesize of a layout for a particular application, circuit, etc., to somesmaller layout units so that each layout unit or a combination ofsmaller layout units, rather than the layout for a whole circuitapplication, may be run with a software package. Using a smaller or agroup of fewer layout units tremendously reduces memory usage. Forexample, without guard banding, e.g., using the whole layout for aparticular circuit application, the coloring software colors allpolygons at the same time in which the memory usage for the systemrunning the software package is invoked for the whole circuitapplication size. In some situation where the layout for the wholecircuit application is used, memory may take up to 30 Gb. In contrast,with guard banding, e.g., using a smaller layout unit, the coloringsoftware colors polygons for that smaller layout unit, and the memorysize is invoked only for that particular smaller layout unit. The sizeof a layout unit varies and depends on design choices by the layoutengineer. For example, the engineer may choose a layout unit to includean independent circuit with a particular function (e.g., processing,storage, input-output, etc.), or circuits with interrelated functions(e.g., storage and input-output, processing and storage, processing andinput-output, etc.), circuits that do not require splitting of thelayout, etc. In some embodiments, the designer may select an arbitrarysize of a layout unit for a particular memory availability of the systemrunning the software. FIG. 6 shows a layout 600 illustrating guardbanding or boxing in accordance with an embodiment. FIG. 6 shows that aparticular layout 600 includes individual layout units L (e.g., L1, L2,L3, . . . LN) wherein, in some embodiments each layout unit L1, L2, L3,. . . LN, etc., is of different shapes and sizes. Further, a layout unit(e.g., layout unit L4′) may include more than one other layout unitincluding layout units L4 and L5, etc. In an embodiment, the customersaves the guard band information after checking/fixing conflict cycles(e.g., in block 115) and the decomposer in block 130 uses thisinformation to save memory usage. For example, the decomposer based onthe layer numbers of each layout unit or guard bands, separates thisunit from other units. Further, when coloring is invoked the coloringtool (e.g., software) colors layout units that have the same layernumbers at a time or that are enclosed by the same guard band.

Layout Decomposition

FIG. 7 shows a flow chart 700 illustrating a method embodiment fordecomposing a layout (e.g., block 130 in FIG. 1). In this illustration,the customer provided a layout to be decomposed by the foundry. Forfurther illustration, the layout has been verified to be split-able inaccordance with techniques described in this document.

In block 605 the layout provided by the customer is used as input to thelayout decomposition software (e.g., the decomposer). The layout, in anembodiment, is in a database file format, e.g., Graphic Data System (GDSII), Open Artwork System Interchange Standard (OASIS), etc.

In block 610 the decomposer applies the layout split rules against theinput layout. In block 615 the decomposer determines whether splittingis necessary. If splitting is not necessary then the decomposer in block620 considers various methodologies to assign the masks, including,random assignment, model-based-assignment, etc.

In block 622, the decomposer assigns layout masks to each polygon. Sincea mask corresponds to a layout (e.g., different mask, different layer),this step may be referred to as assigning layers. In block 625 thedecomposer provides the output layout or tape-out.

In block 615, if, however, splitting is necessary, the decomposer, inblock 630, determines whether polygon cutting is allowed. If polygoncutting is not allowed, then the decomposer in block 660 identifies theconflict polygons and in block 665 identifies conflict edges andcontinues from this block 665 as explained below.

If, however, polygon cutting is necessary and allowed in block 630, thedecomposer in block 635 identifies conflict and connection areas.

In block 640, the decomposer determines if stitching locations areavailable (e.g., previously saved to a file or layers of output layoutin block 320 through pre-stitching or VIRET). If such stitchinglocations are not available, then the decomposer in block 645 performspre-stitching on connection areas. If, however, pre-stitching locationswere previously saved to a file and are now available to the decomposer,the decomposer uses those locations and in block 650 determines ifpre-stitching of a location is feasible. If in block 650 pre-stitchingis not feasible, then the decomposer in block 655 merges connectionareas separated by the potential stitching location but stitching isfound not to be feasible.

The decomposer in block 665 identifies conflict edges and in block 670colors conflict polygons/areas.

The decomposer in block 675 determines if odd cycles should be checked.If there is no need to check odd cycles then the decomposer in block 680determines if manipulating the corner shape (e.g., the right angle in amask) is necessary and/or desirable. If manipulating the corner shape isnecessary and/or desirable, the decomposer in block 682 analyzes thepre-decomposed masks and/or the contours to select a stitching locationbased on a stitching element that will reduce or retain the cornershapes compared to the original layout without decomposition. In anembodiment, the decomposer identifies and/or marks the corner locationof the pre-decomposed layout using geometry recognition from a softwarepackage and selects an appropriate size and shape of a stitch element.The decomposer then generates the pre-decomposed mask/layout patterns,checks the corner number and locations of the pre-decomposed patterns,and compares these patterns with the original layout. The decomposerfinally optimizes (e.g., reduces/increases) the size of the stitchelement based on user requirements that may retain or reduce the numberof corner shapes. In various embodiments of the invention, FEOL(front-end of line) generally uses less corner roundings (e.g., curve ina wafer) and thus reduces them accordingly. In contrast, BEOL (back-endof line) uses more corner roundings, and retains them as appropriate.

If there is no need to manipulate the corner shape then in block 685 thedecomposer finalizes the stitching locations, e.g., taking account oflocation candidates that have been removed, etc., and the method flowsto blocks 622 and 625 to assign the masks and output the layout.

If, in block 675 however, the decomposer determines that odd cycleshould be checked, then the decomposer in block 690 determines if thethere is any odd cycle in the layout. If there is not any odd cycle inthe layout, then the method flows to block 680 and continues therefromas explained above. If, however, there is any odd cycle, then thedecomposer in block 695 determines if it is acceptable to keep the oddcycle. If it is acceptable to keep the odd cycle then the method flowsto block 680 and continues as above.

If in block 695, however, the decomposer determines it is not acceptableto keep the odd cycle then the decomposer in block 698 invokes the fixerto fix the odd cycle. Alternatively or in addition to fixing the oddcycle, the decomposer issues a message to inform the customer that thereis an odd cycle in the layout provided by the customer.

Coloring Conflict Polygons/Areas

Double patterning is a method for decomposing a layout so thatsub-resolution configurations are split into two distinct masks. Thelayout in such situations may be referred to as 2-colorable orbipartite. According to graphic theory, a graph is bipartite or2-colorable if and only if it does not contain an odd cycle (e.g., oddnumber or vertices or edges). A cycle graph is a graph that consists ofa single cycle, or the number of vertices is connected in a closedchain. In various embodiments of the invention a conflict edge connectsa conflict polygon in a first mask (e.g., mask A) to another conflictpolygon in a second mask (e.g., mask B). Embodiments then color thosetwo conflict polygons using a breath-first search (BFS), a graphicsearch, or a coloring software tool that can recognize an odd cycle, ifit exists. In situations where polygon cutting is necessary and/ordesirable to resolve odd cycles, various embodiments of the inventionuse pattern matching or recognition to identify locations candidates forstitching and thus conflict areas (or conflict cycles). Embodiments thenimitate the 2-coloring method to check for odd cycles. As a result,embodiments are advantageous over other approaches that cannot check forodd cycles before splitting the mask.

Merging and Stitch Optimization

In various embodiments of the invention, coloring provides informationfor stitches to be merged and/or polygon(s) to be cut. Generally,determining stitch elements is based on performance (e.g., to keep lesscorner shape in decomposed masks for FEOL but more corner shapes forBEOL). Coloring conflict areas or polygons also provides color sets. Inan embodiment the color relationship between polygons within a color setis identifiable because it is based on a conflict edge that connects aconflict polygon to another conflict polygon. Further, if a conflictpolygon is colored with a first index, e.g., “1,” the other polygonwould be colored with a second index, e.g., “−1.” Additionally, eachcolor set can be assigned a flipping index (e.g., 1 or −1), and can beused to check whether a pre-stitch element connects the polygons of thesame color index (e.g., same color). In that situation, adjoiningconflict polygons or conflict areas separated by the pre-stitch elementsare potentially to be merged. When the pre-stitch element connectsconflict areas of different colors (e.g., different indices), then thecorresponding polygons are potentially to be cut and this pre-stitchelement can be used as a stitch element. In fact, all pre-stitchelements can be stitch elements but the amount of polygon cuttings canbe huge. Embodiments of the invention merge color sets through stitchingelements and use color flipping to minimize the number of polygoncuttings and thus optimize cost.

In one embodiment of the invention, based on the various parameters Fcalculated from the equation,

${F\left( {f_{1},f_{2},{\ldots \mspace{14mu} f_{N}}} \right)} = {\frac{1}{2}{\sum\limits_{i,j}^{N}{f_{i}f_{j}A_{ij}}}}$

determines the parameter F corresponding to a configuration having theoptimized stitch elements or the least number of polygons to be cut.Depending on how the merge/cut index MC (below) is defined, a maximum orminimum value of all possible values (e.g., F_(max)/F_(min)) isconsidered.

In the above equation N is the number of color sets, i< >j.

A_(ij)=A_(ji)

f_(i) is the flipping index of a color set A_(i), which is initially setto 1, and changes to −1 after colors corresponding to the color setA_(i) are flipped,

If the MC index A_(ij) is defined as the number M of stitches thatshould be merged (e.g., potentially be merged) minus the number CT ofstitches corresponding to polygons potentially to be cut whenf_(i)=f_(j)=1, then embodiments determine F_(max) where

${F\mspace{14mu} \max} = {{Max}\left\{ {{F\left( {f_{1},f_{2},{\ldots \mspace{14mu} f_{N}}} \right)} = {\frac{1}{2}{\sum\limits_{i,j}^{N}{f_{i}f_{j}A_{ij}}}}} \right\}}$

If, however, the MC index A_(ij) is defined as the number of stitchescorresponding to polygons potentially to be cut CT minus the number ofstitches that can potentially be merged M when f_(i)=f_(j)=1, thenembodiments determines

${F\mspace{14mu} \min} = {{Min}\left\{ {{F\left( {f_{1},f_{2},{\ldots \mspace{14mu} f_{N}}} \right)} = {\frac{1}{2}{\sum\limits_{i,j}^{N}{f_{i}f_{j}A_{ij}}}}} \right\}}$

F_(min) where

The sum of M and CT is the total number of stitch elements between twocolor sets. Because matrix solver takes all stitch elements into accountin coloring the memory consumption can be very large. Embodiments of theinvention simplify the relationship to an index value and thus reducememory consumption.

f_(i) f_(j)A_(ij) may be referred to as a path from color set A_(i) tocolor set A_(j)

In the below illustrations, the MC index is defined as the number ofstitches that can be merged minus the number of stitches correspondingto polygons to be cut when f_(i)=f_(j)=1. As a result, embodimentsdetermine F_(max) to correspond to a color set configuration having theleast number of polygons to be cut.

FIG. 8 shows graphs 800A-D illustrating minimizing stitch elements withtwo exemplary color sets, in accordance with an embodiment. FIGS. 800A-Dinclude color sets A₁ and A₂ with three stitch areas represented bylines 810, 820, and 830. Further, based on the two color sets, the fourpossible values for parameter F represented by F(f₁, f₂) are F(1,1),F(−1,1), F(1,−1) and F(−1,−1). For illustration, based on these twocolor sets A₁ and A₂, line 820 in graph 800A connects two polygons ofthe same color co1, and therefore these polygons are to be merged. Lines810 and 830 connect two polygons of different colors co1 and co2, andthey are therefore to be cut. As a result, the merged number M is 1 andthe cut number CT is 2, and

A₁₂=A₂₁=M−CT=1−2=−1

Based on the above equation (1)

$\begin{matrix}{{F\left( {f_{1},f_{2}} \right)} = {F\left( {1,1} \right)}} \\{= {\frac{1}{2}\left( {\left( {f_{1}*f_{2}*A_{12}} \right) + \left( {f_{1}*f_{2}*A_{21}} \right)} \right)}} \\{= {\frac{1}{2}\left( \left( {{{1*1*\left( {- 1} \right)} + \left( {1*1*\left( {- 1} \right)} \right)} = {- 1}} \right. \right.}}\end{matrix}$

Graph 800B results from graph 800A with colors for color set A₁ havingbeen flipped (e.g., color co1 becomes color co2 and color co2 becomescolor co1). After flipping, f₁=−1; f₂ remains to be 1; line 810 connectsthe same two colors co1 representing polygons to be merged, and line 830connects the same two colors co2, also representing polygons to bemerged. Line 820 connects two different colors, e.g., color co2 forcolor set A₁ and color co1 for color set A₂, representing polygons to becut. As a result, the merged number M is 2 and the cut number CT is 1,and

$\begin{matrix}{{F\left( {f_{1},f_{2}} \right)} = {F\left( {{- 1},1} \right)}} \\{= {\frac{1}{2}\left( {\left( {f_{1}*f_{2}*A_{12}} \right) + \left( {f_{1}*f_{2}*A_{21}} \right)} \right)}} \\{\left. {= {{\frac{1}{2}\left( {\left( {- 1} \right)*1*\left( {- 1} \right)} \right)} + {\left( {- 1} \right)*(1)\left( {- 1} \right)}}} \right) = 1}\end{matrix}$

Graph 800C results from graph 800A with colors for color set A₂ havingbeen flipped. After flipping, f₁ remains to be 1; f₂=−1; line 810connects the same two colors co2 representing polygons to be merged, andline 830 connects the same two colors co1, also representing polygons tobe merged. Line 820 connects two different colors co1 for color set A₁and color co2 for color set A₂, representing polygons to be cut. As aresult, the merged number M is 2 and the cut number CT is 1, and

$\begin{matrix}{{F\left( {f_{1},f_{2}} \right)} = {F\left( {1,{- 1}} \right)}} \\{= {\frac{1}{2}\left( {\left( {f_{1}*f_{2}*A_{12}} \right) + \left( {f_{1}*f_{2}*A_{21}} \right)} \right)}} \\{= {{{\frac{1}{2}\left( {1*\left( {- 1} \right)*\left( {- 1} \right)} \right)} + \left( {1*\left( {- 1} \right)*\left( {- 1} \right)} \right)} = 1}}\end{matrix}$

Graph 800D results from graph 800A with colors for color sets A₁ and A₂having been flipped. After flipping, f₁=−1, and f₂=−1; line 820 connectsthe same two colors co2 representing polygons to be merged. Line 810 and830 connect two polygons of different colors co1 and co2, and they aretherefore to be cut. As a result, the merged number M is 1 and the cutnumber CT is 2, and

$\begin{matrix}{{F\left( {f_{1},f_{2}} \right)} = {F\left( {{- 1},{- 1}} \right)}} \\{= {\frac{1}{2}\left( {\left( {f_{1}*f_{2}*A_{12}} \right) + \left( {f_{1}*f_{2}*A_{21}} \right)} \right)}} \\{= {{{\frac{1}{2}\left( {\left( {- 1} \right)*\left( {- 1} \right)*\left( {- 1} \right)} \right)} + \left( {\left( {- 1} \right)*\left( {- 1} \right)*\left( {- 1} \right)} \right)} = {- 1}}}\end{matrix}$

Based on the four values F(1,1)=−1, F(−1, 1)=1, F(1,−1)=1, and F(−1,−1)=−1, because the maximum value is equally the same of 1 for bothgraphs 800B and 800C, the stitch optimized solution according toembodiments of the invention could be the configuration corresponding toeither graph 800B or 800C. In both situations the number of polygoncutting is one (CT=1), instead of two (CT=2) as in graph 800A. In fact,the polygons to be merged or to be cut are the same in graphs 800B and800C or graphs 800A and 880D.

FIG. 9 shows graph 900 illustrating stitch optimization having anexemplary four color sets, in accordance with an embodiment of theinvention. In this illustration, as shown in graph 900, the four colorsets include A₁, A₂, A₃, and A₄, and, for illustration purposes, theirMC indices A_(ij) are shown as A₁₂=3, A₁₃=−5, A₁₄=−2, A₂₄=4, A₃₄=2.Further, A₂₃=A₃₂=0 because A₂ does not have a direct connection with A₃.Embodiments of the invention then determine the maximum parameterF_(max) among the various values F of F(f₁,f₂,f₃,f₄) where

$\begin{matrix}{{F\left( {f_{1},f_{2},f_{3},f_{4}} \right)} = {{f_{1}f_{2}A_{12}} + {f_{1}f_{3}A_{13}} + {f_{1}f_{4}A_{14}} + {f_{2}f_{4}A_{24}} + {f_{3}f_{4}A_{34}}}} \\{= {{3\; f_{1}f_{2}} + {\left( {- 5} \right)f_{1}f_{3}} + {\left( {- 2} \right)f_{1}f_{4}} + {4\; f_{2}f_{4}} + {2\; f_{3}f_{4}}}}\end{matrix}$

The total number of possible candidates is 8 and listed as:

F1=F(1,1,1,1), the original configuration as shown in FIG. 9

F2=F(1,1,1,−1), the configuration with colors in color set A4 beingflipped

F3=F(1,1,−1,−1), the configuration with colors in color sets A3 and A4being flipped

F4=F(1,−1, −1,−1), the configuration with color in color sets A2, A3,and A4 being flipped

F5=F(1,−1,1,−1), the configuration with colors in color sets A2 and A4being flipped

F6=F(1,1,−1,1), the configuration with colors in color set A3 beingflipped

F7=F(1,−1,1,1), the configuration with colors in color set A2 beingflipped

F8=F(−1,1,1,−1), the configuration with colors in color sets A1 and A4being flipped

Those skilled in the art will recognize that

F1=F(1,1,1,1)=F(−1,−1,−1,−1)

F2=F(1,1,1,−1)=F(−1,−1,−1,1)

F3=F(1,1,−1,−1)=F(−1,−1,1,1)

F4=F(1,−1, −1,−1)=F(−1,1,1,1)

F5=F(1,−1,1,−1)=F(−1,1,−1,1)

F6=F(1,1,−1,1)=F(−1,−1,1,−1)

F7=F(1,−1,1,1)=F(−1,1,−1,−1)

F8=F(−1,1,1,−1)=F(1,−1,−1,1)

Using the above equation (1)

F1=A₁₂+A₁₃+A₁₄+A₂₄+A₃₄=3+(−5)+(−2)+4+2=2

F2=A₁₂+A₁₃−A₁₄−A₂₄−A₃₄=3+(−5)−(−2)−4−2=−6

F3=A₁₂−A₁₃−A₁₄−A₂₄+A₃₄=3−(−5)−(−2)−4+2=8

F4=−A₁₂−A₁₃−A₁₄+A₂₄+A₃₄=−3−(−5)−(−2)+4+2=10

F5=−A₁₂+A₁₃−A₁₄+A₂₄−A₃₄=−3+(−5)−(−2)+4−2=−4

F6=A₁₂−A₁₃+A₁₄+A₂₄−A₃₄=3−(−5)+(−2)+4−2=8

F7=−A₁₂+A₁₃+A₁₄−A₂₄+A₃₄=−3+(−5)+(−2)−4+2=−12

F8=−A₁₂−A₁₃+A₁₄−A₂₄−A₃₄=−3−(−5)+(−2)−4−2=−6

Because F4=10 is the maximum value (e.g., F_(max)=10), the color setconfiguration corresponding to F4, in accordance with the firstembodiment, is the optimized solution with the least number of polygonsto be cut.

FIG. 10 shows a flowchart 1000 illustrating a method to optimize stitchelements or the number of polygons to be cut, in accordance with anembodiment. In this illustration,

$C_{i}{\sum\limits_{j}^{N}\mspace{14mu} {f_{i}f_{j}A_{ij}\mspace{14mu} {for}\mspace{14mu} {color}\mspace{14mu} {set}\mspace{14mu} A_{i}}}$

In block 1005, all C_(i) are calculated with all f_(i)=f_(j)=1.

In block 1010 the minimum C_(min)(=C_(j)) is determined, and if thisC_(min)>=0 then optimization is complete in block 1030.

If, however, C_(min)<0, then, the index f_(j) is reversed in block 1015.

In block 1020, C_(i) and ΣC_(i) are recalculated.

In block 1025, it is determined if ΣC_(i) is saturated, and if so thenoptimization is completed in block 1030. If, however, in block 1025, itis determined that ΣC_(i) is not saturated, the method embodiment flowsto block 1010 and continues therefrom. ΣC_(i) is considered saturatedwhen its value does not change anymore after a predetermined number ofiterations.

Using the example in FIG. 9,

C₁=A₁₂+A₁₃ A₁₄=3+(−2)+(−5)=−4

C₂=A₁₂ A₂₄=3+4=7

C₃=A₁₃ A₃₄=−5+2=−3

C₄=A₁₄+A₂₄+A₃₄<−2+2+4=4

Alternatively expressing, F(1,1,1,1) results in (C₁, C₂, C₃,C₄)=(−4,7,−3,4). Because C_(min) corresponds to C₁=−4, which is anegative number, embodiments of the invention flip color set A₁ andrecalculate F(−1,1,1,1)

C₁=−A₁₂−A₁₃−A₁₄=−3−(−2)−(−5)=4

C₂=−A₁₂+A₂₄=−3+4=1

C₃=−A₁₃+A₃₄=−(−5)+2=7

C₄=−A₁₄+A₂₄+A₃₄=−(−2)+2+4=8

Consequently F(−1,1,1,1) results in (C₁, C₂, C₃, C₄)=(4,1,7,8)). In thissituation, because C_(min) is 1, which is a positive number, stitchoptimization is complete.

In various embodiments of the invention a stitch may be assigned aweighting value having a default value 1, and the larger the positivevalue of the weighting value, the higher preference for the polygonsassociated with the stitch are to be merged. For a negative weightingvalue, the larger the absolute value of the weighting value, the higherpreference is for the polygons to be cut.

A_(ij) may be defined as

$A_{ij} = {\sum\limits_{s = 1}^{T}{W_{s}P_{i,s}P_{j,s}}}$

where

T is the number of common stitch elements between color set A_(i) andcolor set A_(j)

W_(s) is the weighting value of the s-th stitch element. A weightingvalue W_(s) may be assigned based on one or a combination of the stitchshape (e.g., I, L, T shapes, etc.), the location of stitch elements, thesymmetrical patterns, the user's defined choices, etc. For example,generally, T and L shaped polygons are preferred to be cut while Ishaped polygons are preferred to be merged. Polygons closer to a gateare preferred to be merged, but not cut. When there is a symmetricalpattern and/or symmetrical stitches, if one side is merged, the otherside is also preferred to be merged, but if one side is cut the otherside is preferred to be cut.

P_(i,s) is the color index of polygon in color set A_(i) connected withthe s-th stitch element when f_(i)=f_(j)=1.

P_(j,s) is the color index of polygon in color set A_(j) connected withthe s-th stitch element when f_(i)=f_(j)=1.

In various embodiments, the color index of a polygon, e.g., P_(i,s)and/or P_(j,s) may be represented by 1 or −1. However, the final colorof a polygon is determined by color index and flipping index.

If P_(i,s)=P_(j,s) when f_(i)=f_(j)=1 then the two polygons having thesame color are to be merged, but if P_(i,s)≠P_(j,s) then the twopolygons having the different colors are to be cut.

In the example of FIG. 8A, T=3 (represented by three lines 810, 820, and830). For illustration purposes, W₁₌₁ (line 810), W₂=10 (line 820), andW₃=1 (line 830). Here, because the weighting W₂=10, polygons associatedwith line 820 are preferred to be merged, as compared to other lines 810and 830 each having a weighting value of 1. For further illustration,color co2 is 1, and, as a result color co1 is 1, and P_(1,1)=1,P_(1,2)=−1, and P_(1,3)=−1, P_(2,1)=−1, P_(2,2)=−1, and P_(2,3)=1.

Consequently

$\begin{matrix}{A_{ij} = A_{12}} \\{= {\left( {W_{1}*P_{1,1}*P_{2,1}} \right) + \left( {W_{2}*P_{1,2}*P_{2,2}} \right) + \left( {W_{3}*P_{1,3}*P_{2,3}} \right)}} \\{= {\left( {1*1*\left( {- 1} \right)} \right) + \left( {10*\left( {- 1} \right)*\left( {- 1} \right)} \right) + \left( {1*\left( {- 1} \right)*1} \right)}} \\{= {{- 1} + 10 - 1}} \\{= 8}\end{matrix}$

A_(ij) then may be used in relevant equations as appropriate.

Depending on applications, stitch elements may be associated with morethan 2 color sets (e.g., 3, 4, 5, color sets, etc.), various embodimentsof the invention color polygons associated with the least number ofcolor sets first. For example, in some embodiments, color polygons areassociated with 2 color sets, then color polygons associated with 3color sets, then color polygons associated with 4 color sets, etc.During the coloring process, embodiments reduce the number of color setsas appropriate.

FIG. 11 shows graphs 1100A, 1100B and 1100C, illustrating merging colorsets, in accordance with an embodiment. Generally, merging color sets toreduce/minimize the number of color sets may be used when the number ofcolor sets is large, computing power is limited (e.g., it may take toolong for a computer to optimize stitches in accordance with embodimentsof the invention, etc.).

Graph 1100A shows exemplary seven color sets A₁ to A₇. Color sets A₁ toA₇ are “terminal” color sets because they interact with only one othercolor set. For example, color set A₁ interacts with only color set A₂while color set A₇ interacts with only color set A₆. Embodiments of theinvention merge a terminal color set with the color set interacting withthat terminal color set, e.g., merge color set A₁ with color set A₂, andmerge color set A₇ with color set A₆.

Graph 1100B results from graph 1100A after terminal color set A₁ hasbeen merged with color set A₂ to become color set A₈ and color set A₇has been merged with color set A₆ to become color set A₉ Based on graph1100B, the new color set A₈ is in fact a terminal color set because itinteracts with only one color set A₃. In some embodiments of theinvention, color set A₈ is further merged with color set A₃.

Graph 1100C results from graph 1100B after terminal color set A₈ hasbeen merged with color set A₃ to become color set A₁₀ while other colorsets A₄, A₅, and A₉ remain unchanged. As compared to graph 1100A thatincludes 7 color sets, in some embodiments of the invention, the numberof color sets is reduced to 5 color sets in graph 1100B and 4 color setsin graph 1100C.

BACKGROUND INFORMATION AND TERMS

FIG. 12 shows graphs illustrating terms used in this document. Asbackground information for a better understanding of the invention, thefollowing terms and their definitions are used, but they are commonlyrecognizable and understandable by a person skilled in the art.

Polygon 1210 and 1225: represents a circuit and includes edges andvertices.

Conflict space 1215: a region created nearby polygons' edges and/orvertices to determine conflicts (e.g., space/pitch conflicts).

Conflict area 1220: the overlap between the polygon and the conflictspace.

Conflict polygons 1225: polygons that violate the spacing and/orsplitting rules (e.g., two polygons that are separated by a spacesmaller than a predetermined value).

Conflict edge 1230: a segment connecting a pair of conflict polygons orareas.

Connection area 1235: the remainder part of a polygon excluding theconflict area.

Stitch area 1240: an area from the connection area that separatesconflict areas. The conflict space, conflict area, connection area andstitch area created by polygon 1220 are not shown in FIG. 12 to notobscure the drawing.

The above method embodiments of the invention may be implemented insoftware or instructions executed by a computer. These instructions maybe stored in and/or carried through one or more computer-readable media,which refers to any medium from which a computer reads information(e.g., a CD-ROM, a DVD-ROM, optical medium, etc.). For example, theinstructions may be in the form of executable binary stored in a CD-ROMinterfaced with the computer system. The computer system then loadsthese instructions in RAMs, executes some instructions, and sends someinstructions via a communication interface (e.g., a modem) and atelephone line to a network, the Internet, etc., and stores the data ina storage device, e.g., memories, etc. The computer system may alsodisplay data (e.g., the conflicting polygons, the conflicting edges, theodd cycles, exemplary polygons to be used/avoided, etc.) on its screen,etc.

A number of embodiments of the invention have been described. It willnevertheless be understood that various modifications may be madewithout departing from the spirit and scope of the invention. Forexample, embodiments of the invention can be used in multiple exposuresystem in which a trim mask is used to form the end-end pattern afterdouble patterning. The Specification above describes method embodimentsto determine the least number of polygons to be cut. Generally, theembodiment using the equation (1) is used for a smaller number of colorsets, and the embodiment depicted in FIG. 10 is used for a large numberof color sets. However, embodiments of the invention are not so limited.Selecting one embodiment over the other embodiment can depend on otherfactors, including for example the available computing power and/orwhether the number of color sets can be reduced (e.g., illustrated FIG.11). For example, when the computing power is limited, a smaller colorsets can still take a long time, the embodiment in flowchart 1000 may beused in that situation. In contrast, if the available computing power ispowerful, some embodiments applying equation (1) may still be used for alarge number of color sets. Further, the initial number of color set maybe large, but it may be reduced and considered as a smaller number ofcolor sets, etc. The above method embodiments show exemplary steps, butthese steps are not necessarily required to be performed in the ordershown. Steps may be added, replaced, changed order, and/or eliminated asappropriate, in accordance with the spirit and scope of embodiments ofthe invention.

Each claim of this document constitutes a separate embodiment, andembodiments that combine different claims and/or different embodimentsare within scope of the invention and will be apparent to those skilledin the art after reviewing this disclosure. Accordingly, the scope ofthe invention should be determined with reference to the followingclaims, along with the full scope of equivalences to which such claimsare entitled.

1. A method comprising: generating polygons representing circuits of anintegrated circuit; identifying one or a combination of a conflictpolygon, a conflict edge, or an odd cycle; and if the odd cycle existsthen eliminating the odd cycle.
 2. The method of claim 1 wherein theidentifying is performed while the polygons are being generated.
 3. Themethod of claim 1 further providing polygons that have been verified tobe split-able and polygons that have been verified to be un-split-ablefor use in generating the polygons corresponding to the circuits of theintegrated circuit.
 4. The method of claim 1 further identifying aconflict area and a connection area if cutting a polygon is allowable.5. The method of claim 4 further identifying a stitching locationcandidate.
 6. The method of claim 5 wherein if the stitching locationcandidate is not used for stitching, further merging polygons beingseparated by the stitching location candidate.
 7. The method of claim 1wherein eliminating the odd cycle is done by using a verticalintegration resolution enhancement technology.
 8. The method of claim 1wherein eliminating the odd cycle is done by revising at least onepolygon.
 9. The method of claim 1 further limiting a size of a layoutcorresponding to the circuits of the integrated circuit.
 10. The methodof claim 9 further saving information related to the size to a file forlater use.
 11. The method of claim 1 further using stitching elements toidentify stitching location candidates.
 12. The method of claim 11further determining if a stitching location candidate can be used forstitching, and if the stitching location candidate cannot be used forstitching, merging polygon portions separated by the stitching locationcandidate.
 13. The method of claim 12 further saving stitching locationsto a file.
 14. The method of claim 13 wherein the saved stitchinglocations are later used in decomposing a layout of the integratedcircuit.
 15. The method of claim 1 further changing a number of cornershapes based on a user requirement.
 16. The method of claim 1 furtherchanging a number of corner shapes based on a size of a stitchingelement.
 17. A computer readable-medium having computer instructionsstored therein, the computer instructions performing a methodcomprising: providing polygons representing integrated circuits; using aconflict space to identify at least one conflict area; indicating thatpolygon cutting is allowed; identifying at least one conflict edge basedon the at least one conflict area; if a connection area exists, using astitching element to determine if the connection area is a stitchinglocation candidate; and identifying an odd cycle based on the at leastone conflict edge; and if the odd cycle is identified, resolving the oddcycle.
 18. The computer-readable medium of claim 17 wherein if thestitching location candidate is not used for stitching further mergingpolygon portions separated by the stitching location candidate.
 19. Thecomputer-readable medium of claim 17 wherein if the stitching locationcandidate is to be used for stitching the method further savinginformation related to the stitching location candidate to a file forlater use.
 20. The computer-readable medium of claim 17 wherein themethod further reading stitching locations in a file for decomposing.21. The computer-readable medium of claim 17 wherein the method furtherselecting a stitching location based on a stitching element tomanipulate a number of corner shapes.
 22. The computer-readable mediumof claim 17 wherein the method further extending a margin of a stitchingarea.
 23. The computer-readable medium of claim 17 wherein the methodfurther using guard band information related to a layout unit to savememory usage.
 24. The computer-readable medium of claim 17 wherein themethod further displaying one or a combination of a polygon that can beused and a polygon to be avoided on a display.
 25. A computer-readablemedium having computer instructions stored therein, the computerinstructions performing a method comprising: displaying one or acombination of a polygon that can be used or a polygon that should notbe used; providing options to select whether polygon cutting is allowedor not; displaying polygons corresponding to integrated circuits whenthe polygons are created; displaying at least one conflict polygon whenthe at least one conflict polygon is created; displaying at least oneconnection edge connecting the least one conflict polygon if the atleast one conflict polygon is created; and indicating that an odd cycleexists if the odd cycle is created.
 26. A method comprising: providing Ncolor sets used in coloring polygons a layout of an integrated circuitand represented by A₁ to A_(N); corresponding a set of indices f₁ tof_(N) to the N color sets A₁ to A_(N); wherein an index f_(i)corresponds to a color set A_(i); determining a number of potentialmerges M corresponding to the color set A_(i) and a color set A_(j);where i≠j; determining a number of potential cuts CT corresponding tothe color set A_(i) and the second color set A_(j); referring thedifference between the number of potential merges M and the number ofpotential cuts CT as an index A_(ij) where A_(ij)=A_(ji); obtaining aplurality of parameters F related to the index A_(ij); and selecting aparameter F among the plurality of parameters F based on a definition ofthe index A_(ij).
 27. The method of claim 26 wherein the index f_(i)includes a value of 1 or −1.
 28. The method of claim 26 whereinselecting the parameter F among the plurality of parameters F having ahighest value if the index A_(ij) is defined as the number of potentialmerges M minus the number of potential cuts CT and selecting theparameter F among the plurality of parameters F having a lowest value ifthe index A_(ij) is defined as the number of potential cuts CT minus thenumber of potential merges M.
 29. The method of claim 26 whereinobtaining the plurality of parameters F is based on the equation${F\left( {f_{1},f_{2},{\ldots \mspace{14mu} f_{N}}} \right)} = {\frac{1}{2}{\sum\limits_{i,j}^{N}\; {f_{i}f_{j}A_{ij}}}}$30. The method of claim 26 further merging a terminal color set to acolor set that interacts with the terminal color set.
 31. The method ofclaim 26 wherein the index A_(ij) includes a weighting value based onone or a combination of a stitch shape, a location of stitch elements, asymmetrical pattern, a user's defined choice.
 32. The method of claim 26wherein the index A_(ij) is represented by$A_{ij} = {\sum\limits_{s = 1}^{T}\; {W_{s}P_{i,s}P_{j,s}}}$ where Tis a number of common stitch elements between the color set A_(i) andthe color set A_(j) W_(s) is a weighting value of an s-th stitchelement. P_(i,s) is a color index of polygon in the color set A_(i)connected with the s-th stitch element; and P_(j,s) is a color index ofpolygon in the color set A_(j) connected with the s-th stitch element.33. The method of claim 32 wherein f_(i) and f_(j) are assigned
 1. 34. Amethod comprising: providing N color sets used in coloring polygons alayout of an integrated circuit and represented by A₁ to A_(N);corresponding a set of indices f₁ to f_(N) to the N color sets A₁ toA_(N); whereby an index f_(i) corresponding to a color set A_(i);determining a number of potential merges M corresponding to the colorset A_(i) and a color set A_(j); where i≠j; determining a number ofpotential cuts CT corresponding to the color set A_(i) and the secondcolor set A_(j); referring the difference between the number ofpotential merges M and the number of potential cuts CT as an indexA_(ij) where A_(ij)=A_(ji); based on the equation$C_{i} = {\sum\limits_{j}^{N}\; {f_{i}f_{j}A_{ij}}}$ for the colorset A_(i) calculating a plurality of C_(i) wherein the index f_(i)corresponds to a C_(i) is a positive number; and based on the pluralityof C_(i) determining a color set configuration corresponding to a leastnumber of polygons to be cut.
 35. The method of claim 34 furthercomprising determining a C_(j) among the plurality of C_(i) having alowest value; if C_(j)<0, then flipping an index f_(j) to a negativenumber; and recalculating$C_{i} = {\sum\limits_{j}^{N}\; {f_{i}f_{j}A_{ij}}}$
 36. The methodof claim 35 where the positive number is equal to 1 and the negativenumber is equal to −1.
 37. The method of claim 34 further merging aterminal color set to color set interacting with the terminal color set.38. The method of claim 34 wherein the index A_(ij) includes a weightingvalue based on one or a combination of a stitch shape, a location ofstitch elements, a symmetrical pattern, a user's defined choice.
 39. Themethod of claim 38 wherein the index A_(ij) is represented by$A_{ij} = {\sum\limits_{s = 1}^{T}\; {W_{s}P_{i,s}P_{j,s}}}$ where Tis a number of common stitch elements between the color set A_(i) andthe color set A_(j) W_(s) is a weighting value of an s-th stitchelement. P_(i,s) is a color index of polygon in the color set A_(i)connected with the s-th stitch element; and P_(j,s) is a color index ofpolygon in the color set A_(j) connected with the s-th stitch element.40. The method of claim 32 wherein f_(i) and f_(j) are assigned 1.